Efficient algorithms for verified scientific computing : Numerical linear algebra using interval arithmetic

Hong Diep Nguyen 1, 2
2 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Interval arithmetic is a means to compute verified results. However, a naive use of interval arithmetic does not provide accurate enclosures of the exact results. Moreover, interval arithmetic computations can be time-consuming. We propose several accurate algorithms and efficient implementations in verified linear algebra using interval arithmetic. Two fundamental problems are addressed, namely the multiplication of interval matrices and the verification of a floating-point solution of a linear system. For the first problem, we propose two algorithms which offer new tradeoffs between speed and accuracy. For the second problem, which is the verification of the solution of a linear system, our main contributions are twofold. First, we introduce a relaxation technique, which reduces drastically the execution time of the algorithm. Second, we propose to use extended precision for few, well-chosen parts of the computations, to gain accuracy without losing much in term of execution time.
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Theses
Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2011. English. < NNT : 2011ENSL0617 >
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Hong Diep Nguyen. Efficient algorithms for verified scientific computing : Numerical linear algebra using interval arithmetic. Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2011. English. < NNT : 2011ENSL0617 >. <tel-00680352>

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